From a Buddhist point of view, dogs, cats, companion animals — in fact all animals and sentient life — are equally precious. According to Buddha Dharma, all sentient beings also have Buddha Nature — the potential to eventually become enlightened.
Moreover, the doctrine of rebirth held that any human could be reborn as animal, and any animal could be reborn as a human. An animal might be a reborn dead relative, and anybody who looked far enough back through their series of lives might come to believe every animal to be a distant relative.
The Buddha expounded that sentient beings currently living in the animal realm have been our mothers, brothers, sisters, fathers, children, friends in past rebirths. Ultimately humans and animals were part of a single family. They are all interconnected.
Rebirth as an animal was considered to be one of the unhappy rebirths, usually involving more than human suffering.
Lobsang Dhargey, resident teacher at the Tibetan Buddhist Centre in Redding advises “recite mantras and pray for the animals.” Not when they are dying, but throughout their lives.
Kyabje Lama Zopa Rinpoche also advises these mantras or practices as beneficial for animals, throughout their lives to help bless their mindstreams:
Medicine Buddha: Om Bekhandze Bekhandze Maha Bekhandze Bekhandze Randza Samundgate Soha
Green Tara: Om Tare Tuttare Ture Soha
Avalokiteshvara (Chenrezig or Guanyin): Om Mani Padme Hum
Milarepa: Om Ah Guru Hasa Vajra Sarva Siddhi Hum
In the case of a deceased companion, the same mantras are beneficial, chanted at least 21 times, preferably 108, blowing water and visualizing the deity or the deity’s blessing light being absorbed. Then pour the water on the deceased.
For a dying animal, Lama Zopa Rinpoche recommended saying the entire Medicine Buddha sadhana.
In Thailand, where the population is mostly Buddhist, euthanasia is unthinakable. For this reason, thousands of dogs are taken to temples to die in peace under the care of monks. They know monks consider life precious and will share food— this despite the fact these monks depend on alms themselves to live.